Questions regarding functions defined recursively, such as the Fibonacci sequence.

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5
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3answers
131 views

The number of length-n ternary sequences with even ones and even zeroes

Just starting to appreciate recurrence relations Let $T_n = $ number of length-n ternary sequences with an even number of ones and an even number of zeroes. $T_0 = 1$, because $0$ is an even number, ...
5
votes
2answers
117 views

Why should we suspect that the recurrence $T(n) = T(n-1) + n(n-1)$ satisfies a polynomial identity?

In the question Algorithms: Recurence Relation, the author asked about the recurrence relation $$T(n) = T(n-1) + n(n-1)$$ and one of the answers proposed assuming $T(n)$ is polynomial, then ...
5
votes
1answer
168 views

Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$

I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $ then the general equation looks like the following: $$ F_n = F_{n-1}(F_{n-2}) + F_{n-3} $$ So far I have got the ...
5
votes
4answers
127 views

Recurrence equation: $u_n = 4u_{n−1} + 4u_{n−2}$ ; is $4x+4 = 4$ the characteristic equation?

Given this recurrence equation: $u_1 = 0, u_2 = 1$ $u_n = 4u_{n−1} + 4u_{n−2}$ Is the correct characteristic equation: $4x+4 = 4$ EDIT: Complete solve: The characteristic equation is ...
5
votes
2answers
371 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
5
votes
2answers
154 views

Solving a recurrence for a probability?

I came across the following recurrence relation when exploring properties of a certain type of randomized perfect binary tree: $$ T(0) = \frac{1}{2} $$ $$ T(k + 1) = 1 - T(k)^2 $$ (Specifically, ...
5
votes
1answer
187 views

Finding the general formula $a_n$ for $a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$

How to calculate the general formula $a_n$ for the following sequence: $$a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$$ where $a_1=\frac{1}{2}, a_2=\frac{1}{4}$
5
votes
2answers
247 views

Whats better: 1 million dollars in a month or a penny(USD) doubled (and added) every day for 30 days?

THis is a question that I remember when I was in the 5th grade that tested our logical reasoning skills. And it is a simple choice knowing that the pennies doubling every day is an exponential ...
5
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2answers
155 views

Recurrence relation: composition of a polynomial function

Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is $$f_1(x)=f(x)$$ $$f_2(x)=f \circ f(x)$$ $$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$ Is there a way to find a formula for ...
5
votes
1answer
226 views

When is a recurrence relation linear

In http://www.cs.fsu.edu/~lacher/courses/COT5405/spring07/notes2.html, it says that $T(n) = aT(n/b) + f(n) $ is nonlinear recurrence. But I think it is linear because $T(n)$ is linear in $T(n/b)$. ...
5
votes
3answers
675 views

how to solve $T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}$

How can I solve the recurrence $$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$ Please don't just write the name of the method, as I just started learning this stuff and things are a ...
5
votes
2answers
147 views

Let $x_{n+1} = x_n + 1/(x_1 + x_2 +\ldots + x_n)$ with $x_1 = 1$. Show that $x_n\sim\sqrt{2\log(n)}$.

As the title states we have a sequence defined by $$x_{n+1} = x_n + \dfrac{1}{x_1 + x_2 + \cdots + x_n}$$ with $x_1 = 1$. The first few terms are: $1, 2, \frac{7}{3}, \frac{121}{48} \cdots$ Any ...
5
votes
4answers
3k views

What is the solution to the following recurrence relation with square root?

This looks like a question asked earlier, but it isn't T(n) = T (sqrt(n)) + 1 ... if n>1 =1... if n=1 My professor gave this to me in class yesterday. This is where I'm stuck.. T(n) ...
5
votes
1answer
341 views

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is an exercise from Chapter 1 of "Concrete Mathematics". It concerns the Towers of Hanoi. Are there any starting and ending configurations of $n$ disks on three pegs that are more than $2^n - ...
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2answers
1k views

Second order homogeneous linear difference equation with variable coefficients

I was wondering if you would point me to a book where the theory of second order homogeneous linear difference equation with variable coefficients is discussed. I am having difficulties in getting ...
5
votes
3answers
161 views

Solve the recursion, $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$

Bring the following recursion relation to an explicit expression: $$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$$ $a_{0} = 0$, $a_1 = 1$, $a_2 = 2$ All the examples I have seen were with maximum 2 steps back ...
5
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2answers
63 views

Let $A$ be a matrix sized $p\times p$, where $2\le p$. Using recurrence relations, describe $A^k$.

Let $A$ be a matrix sized $p\times p$, Where $2\le p$. The matrix values in the main diagonal are $0$ and the rest are $1$'s. Example for $A$ where $p=5$: $$\begin{bmatrix} 0 & 1 & 1 & 1 ...
5
votes
2answers
826 views

Solving recurrence relation of form $T(n/2 + c)$

It is obvious that the Master Theorem cannot be applied to the recurrences of the following form: $T(n) = 4T(n/2 + 2) + n$ Since I am only interested in the $\theta$ bound of the recurrence and not ...
5
votes
1answer
190 views

two-dimensional recurrence

Can someone using only these conditions $$a_{m,k}=a_{m-1,k}+a_{m-1,k-1},m>k$$ $$a_{m,k}=1,m=k$$ $$a_{m,k}=0,m<k$$ prove that $$a_{m,k}=\frac{m!}{k!(m-k)!}$$ here is way to construct Pascal ...
5
votes
2answers
423 views

Show that the Catalan numbers are given by this recurrence relation

Hey guys! I'm doing an assignment, and I'm just not sure (at all) how to start this problem. Can somebody nudge/shove me in the right directions? Show that the Catalan numbers are given by the ...
5
votes
1answer
263 views

Recurrence for perfect matchings revisited.

I like to study combinatorics a bit as a hobby, and recently a question I found interesting was posed asking to derive a recurrence for the generating function $G_n(x)$, the ordinary generating ...
5
votes
1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
5
votes
3answers
124 views

How to solve this recurrence $T(n)=2T(n/2)+n/\log n$

How can I solve the recurrence relation $$T(n)=2T\left(\frac n2\right)+\frac{n}{\log n}$$? I am stuck up after few steps.. I arrive till $$T(n) = 2^k T(1) + \sum_{i=0}^{\log(n-1)} ...
5
votes
1answer
271 views

Deriving a recurrence relation

The number of sequences of length $n$ consisting of positive integers such that the opening and ending elements are $1$ or $2$ and the absolute difference between any $2$ consecutive elements is $0$ ...
5
votes
1answer
84 views

Geometric interpretation of the fundamental theorem for coalgebras?

Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the ...
5
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2answers
1k views

Solving recurrence $T(n) = T(\lceil n/2 \rceil) + T(\lfloor n/2 \rfloor) + \Theta(n)$

I'm learning algorithms by myself and am using the excellent Introduction to algorithms to do that. It has been quite a long time since I last studied math, so maybe the solution to my problem is ...
5
votes
1answer
156 views

general technique to convert recurrence relation to integral

I know the following recurrence relation $$a_n=\frac{a+na_{n-1}}{a-n}$$ with $a_0=1$ can be represented alternatively as an integral $$a_n=a\int_0^1{x^{a-n-1}(2-x)^ndx}$$ Verifying this is easy, ...
5
votes
1answer
232 views

A nicer recurrence for the Eulerian polynomials.

I was perusing the subject of Eulerian polynomials. I'm assuming the definition that the Eulerian polynomial is defined by $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of ...
5
votes
1answer
298 views

Purely combinatorial proof and simplification of identity involving factorials and summations

While trying to decompose factorials into summations, I came up with the following identity $$(n+2)! = 2^{n+1} + \sum\limits_{k=0}^{n-1}\sum\limits_{i=0}^{n-1-k}\sum\limits_{S \subseteq ...
5
votes
3answers
303 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
5
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1answer
73 views

Recurrence-differential equation

In his book on differential equations, Arnold writes that $x'(t)=x(x(t))$ is not a differential equation. My question is: how can one solve it?
5
votes
1answer
553 views

Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $

Express recurrence relation of the integral $$ I_n=\int\frac{dx}{(1+x^2)^n} $$ [My Answer] $$ I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx $$ $$ I_n=I_{n-1}-\int ...
5
votes
0answers
81 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...
5
votes
0answers
35 views

Finding a recurrence relation for a sequence not containing $0$ in the digits. Also showing $\sum\limits_{k=1}^n \dfrac{1}{a_k} < 90$

Let $1,2,3,4,5,6,7,8,9,11,12,\ldots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that ...
5
votes
0answers
58 views

Right way to solve this reccurence relation?

I am not that good in mathematics , I have solved many simple relations. But I am not be able to solve the following. This is not a homework. I found it during analysis of a program. ...
5
votes
0answers
74 views

Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
5
votes
0answers
123 views

number of potential couples

A potential couple is a pair of a man and a woman that like each other (assume that 'like' is a symmetric relation). Given a group of $M$ men and $W$ women, I want to know how many different ...
5
votes
0answers
511 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
4
votes
5answers
291 views

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$?

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for ...
4
votes
4answers
314 views

What's known about recurrences involving $(a_n)^2$?

I've run across the recurrence $a_{n+1} = (a_n)^2 + 1$ in the past. Unfortunately, the referrence escapes me. However, my impression was that recurrences involving the product of previous terms ...
4
votes
4answers
212 views

Recurrence relation $C_n = n + 1 + \dfrac{2}{n}\sum\limits_{k=0}^{n-1}C_k$.

A Discrete Mathematics book from which I'm self-studying ("Discrete Mathematics and Its Applications", by Kenneth Rosen) asks me to do the following: Given the following recurrence relation: $$C_n = ...
4
votes
4answers
256 views

Finding the billionth number in the series: $2, 3, 4, 6, 9, 13, 19, 28, 42, \ldots $?

Series is defined as $$a_{n+1} = \lfloor\frac{3\cdot a_n}{2}\rfloor,\qquad a_0 = 2$$ It can be viewed as the number of animals starting from a single pair if any pair of animals can produce a single ...
4
votes
2answers
123 views

Does this problem have a name?

Recently our lecturer told us that it is an unsolved mathematical problem if the following while loop aka iteration ever terminates. Unfortunately I forgot to ask him what it is called. If someone ...
4
votes
2answers
1k views

Fibonacci, tribonacci and other similar sequences

I know the sequence called the Fibonacci sequence; it's defined like: $\begin{align*} F_0&=0\\ F_1&=1\\ F_2&=F_0+F_1\\ &\vdots\\ Fn&=F_{n-1} + F_{n-2}\end{align*}$ And we ...
4
votes
4answers
275 views

Explicit formula for a recursion

How can you express the following recursion explicitly? \begin{cases} T_0 = 1\\ T_n = 1 + 2\cdot T_{n-1}\\ \end{cases}
4
votes
2answers
295 views

What is the closed form of this recurrence relation

I have the following recurrence relation : $$g(0) = c $$ $$g(i+1) = g(i) + (1-g(i))*g(i)^{2}$$ where 0 < c < 1. Is there any closed form for this relation? If not can you give me an upper ...
4
votes
4answers
364 views

How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms?

I know how to solve "simple" recurrence relations. For instance, say you have: $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$ We can write the characteristic equation as: $$3x^{n-1} - ...
4
votes
3answers
963 views

Recurrence relation (linear, second-order, constant coefficients)

Q1. Find the general solution to the difference equation $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$ Q2. Solve the difference equation $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$ I am ...
4
votes
2answers
256 views

Sum of the series formula

I need to figure out the sum of the series as quickly as possible in a program given n and k: $$f(n,k)= ...
4
votes
3answers
2k views

Solving $T(n)= 2T(n/2)+n \lg (n)$

I am trying to solve a recursive function: $$ T(n)= 2 T(n/2)+n \lg(n), \quad n>2,\quad T(2)=2,\quad n = 2^{k}$$ Master theorem didn't work. The result is pointless (if I did it right). Any ...